Differential and Integral Calculus I

Objectives

Master concepts and techniques of differentiable and integral calculus in one variable. Develop analytic thinking, creativity and innovation capacity, through the application of those concepts and techniques in different contexts.

Program

1. Real numbers: algebraic, order and supreme axioms. Natural numbers and mathematical induction.

2. Real functions of a real variable

2.1 limit and continuity

2.2 Bolzano and Weierstrass theorems

3. Differential Calculus in IR

3.1 Derivative at a point and its geometric interpretation

3.2 Derivative function

3.3 Derivative rules

3.4 Rolle, Lagrange and Cauchy theorems; Limits'  Indeterminations

3.5 functions graphs

3.6 Taylor's polynomials.

4. Primitivation

4.1 Definition of primitive

4.2 Primitives: immediate, by parts, of rational functions, and by substitution 

5. Integral calculus in IR.

5.1 Riemann integral and its properties

5.2 Fundamental theorem of calculus and Barrow's formula

5.3 Applications: computation of areas and lengths of curved arches

6. Sequences and series of real numbers

6.1 Monotonous, limited and convergent successions

6.2 Limit of a sequence and fundamental theorems

6.3. Geometric, Geometric-Arithmetic and Mengoli series

6.4 Series convergence criteria.

6.5 Alternate series

6.6 Absolutely and simply convergent series

6.7 Power series: radius, range and convergence domain

Teaching Methodologies

The contents are consistent with the objectives of this course and follow a constructive and dynamic strategy. The topics are intended to provide students with basic techniques and basic knowledge to understand the fundamental concept of limit and proceed with limits, derivatives, and primitive computations essential to the success of this course and others that need these concepts as prerequisites. 

The theoretical and practical analysis of the topics should be buit on students maturity and based on mathematical formalisms whose intention is to familiarize and equip students with knowledge and skills to progress in the process of training and in their future profession. 

Bibliography

Calculus, M. Spivak, 2006, 3rd Edition, Cambridge University Press;

Introduction to Real Analysis, W. Trench, 2009, (free edition) Trinity University;

Introdução à Análise Matemática, J. Campos Ferreira, 2018, 12.ª edição, Gulbenkian;

 A First Course in Real analysis, M. H. Protter e C. B. Morrey, 1993, Springer-Verlag;

Calculus, J. Stewart, 2015, 8th edition.

Aulas teóricas de Cálculo Diferencial e Integral I, M. Abreu e R. L. Fernandes, 201